COHOMOLOGY OF GROUPS : EXERCISES

Sheet 7, 9 July 2007

Exercise 7.1. Let p ⩾ 3 be a prime, let C_{p} = ⟨t|t^{p} = 1⟩ and C_{p^{2}} = ⟨s|s^{p^{2}} = 1⟩ be cyclic groups of order p and p^{2}. We let C_{p} act on C_{p^{2}} by t(s) = s^{1+p}. Let E = C_{p^{2}} ⋊ C_{p} be the corresponding extension

1 → C_{p^{2}} → E → C_{p} → 1.

Compute H^{∗}(E; 𝔽_{p}) as an algebra. Hint: Consider the LHS spectral sequence associated to this extension. Show that it has a non-trivial d_{2} differential by computing H^{2}(E; 𝔽_{p}) and comparing it with E_{2}^{0,2} ⊕ E_{2}^{1,1} ⊕ E_{2}^{2,0}. To compute H^{2}(E; 𝔽_{p}), you can for example use an explicit resolution, or first compute H^{i}(E; ℤ) for i = 2, 3 with the LHS spectral sequence and apply universal coefficients. Show that it collapses at the E_{3}-term. Show that there are no multiplicative extensions.

Exercise 7.2. Compute H^{∗}(D_{8}; 𝔽_{2}) as an 𝔽_{2}-algebra. Hint: Prove that there is an isomorphism D_{8} ≅ C_{2} ≀ C_{2}.

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